If C and D are two events such that `CsubD""a n d""P(D)!=0` , then the correct statement among the following is : (1) `P(C|D)""=""P(C)` (2) `P(C|D)geqP(C)` (3) `P(C|D)""<""P(C)` (4) `P(C|D)""=""P(D)//(P(C)`

Options

A. `P((C)/(D)) ge P (C )`
B. `P((C)/(D)) lt P (C )`
C. `P((C)/(D)) =(P(D))/(P(C))`
D. `P((C)/(D)) =P(C)`

Answer

To solve the problem, we need to analyze the relationship between events C and D, given that C is a subset of D (C ⊆ D) and that the probability of D is not zero (P(D) ≠ 0). We will evaluate the four statements provided to determine which one is correct.

### Step-by-step Solution:

1. **Understanding Conditional Probability**:
We start with the definition of conditional probability:
\[ P(C|D) = \frac{P(C \cap D)}{P(D)} \]
Since C is a subset of D, the intersection of C and D is just C itself:
\[ P(C \cap D) = P(C) \]
Therefore, we can rewrite the conditional probability:
\[ P(C|D) = \frac{P(C)}{P(D)} \]

2. **Analyzing the Statements**:
Now we will analyze each of the four statements given in the problem:

- **Statement (1)**: \( P(C|D) = P(C) \)
This is incorrect because \( P(C|D) = \frac{P(C)}{P(D)} \) and since \( P(D) \neq 0 \), \( P(C|D) \) cannot equal \( P(C) \).

- **Statement (2)**: \( P(C|D) \geq P(C) \)
This is also incorrect because \( P(C|D) = \frac{P(C)}{P(D)} \) implies that \( P(C|D) < P(C) \) since \( P(D) > 1 \) would not hold true.

- **Statement (3)**: \( P(C|D) < P(C) \)
This is correct because \( P(C|D) = \frac{P(C)}{P(D)} \) and since \( P(D) > 0 \), \( P(C|D) \) will always be less than \( P(C) \) as long as \( P(D) > 1 \).

- **Statement (4)**: \( P(C|D) = \frac{P(D)}{P(C)} \)
This is incorrect as it misrepresents the relationship between the probabilities.

3. **Conclusion**:
The correct statement among the options provided is:
\[ \text{(3) } P(C|D) < P(C) \]